3.763 \(\int \frac{1}{(a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=503 \[ -\frac{\left (-2 a^2 b^2 \left (4 c^2-3 d^2\right )+20 a^3 b c d-15 a^4 d^2+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}-\frac{\left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

[Out]

(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (3*b^2*(2*a
*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e +
f*x])) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((6*a*b*c - 7*a^2*d + b^2*d)*Ellipt
icF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt
[c + d*Sin[e + f*x]]) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3
*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a
- b)^2*(a + b)^3*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.67523, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\left (-2 a^2 b^2 \left (4 c^2-3 d^2\right )+20 a^3 b c d-15 a^4 d^2+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}-\frac{\left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (3*b^2*(2*a
*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e +
f*x])) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x
]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((6*a*b*c - 7*a^2*d + b^2*d)*Ellipt
icF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt
[c + d*Sin[e + f*x]]) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3
*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a
- b)^2*(a + b)^3*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}-\frac{\int \frac{\frac{1}{2} \left (-4 a b c+4 a^2 d-3 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+\frac{1}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac{3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac{\int \frac{\frac{1}{4} \left (-16 a^3 b c d-2 a b^3 c d+8 a^4 d^2+a^2 b^2 \left (8 c^2-5 d^2\right )+b^4 \left (4 c^2+3 d^2\right )\right )+\frac{1}{2} b d \left (5 a^2 b c+b^3 c-8 a^3 d+2 a b^2 d\right ) \sin (e+f x)+\frac{3}{4} b^2 d \left (2 a b c-3 a^2 d+b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac{3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac{\int \frac{\frac{1}{4} b d \left (7 a^3 b c d+5 a b^3 c d-8 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right )+\frac{1}{4} b^2 d (b c-a d) \left (6 a b c-7 a^2 d+b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )^2 d (b c-a d)^2}+\frac{\left (3 b \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac{3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac{\left (6 a b c-7 a^2 d+b^2 d\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)}-\frac{\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2}+\frac{\left (3 b \left (2 a b c-3 a^2 d+b^2 d\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac{3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac{3 b \left (2 a b c-3 a^2 d+b^2 d\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (\left (6 a b c-7 a^2 d+b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac{3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac{3 b \left (2 a b c-3 a^2 d+b^2 d\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (6 a b c-7 a^2 d+b^2 d\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{4 \left (a^2-b^2\right )^2 (b c-a d) f \sqrt{c+d \sin (e+f x)}}-\frac{\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 (a+b)^3 (b c-a d)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 7.69447, size = 1069, normalized size = 2.13 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{3 \left (d \cos (e+f x) b^4+2 a c \cos (e+f x) b^3-3 a^2 d \cos (e+f x) b^2\right )}{4 \left (a^2-b^2\right )^2 (a d-b c)^2 (a+b \sin (e+f x))}-\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (a d-b c) (a+b \sin (e+f x))^2}\right )}{f}+\frac{-\frac{2 \left (16 d^2 a^4-32 b c d a^3+16 b^2 c^2 a^2-19 b^2 d^2 a^2+2 b^3 c d a+8 b^4 c^2+9 b^4 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (4 c d b^4+8 a d^2 b^3+20 a^2 c d b^2-32 a^3 d^2 b\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (-3 d^2 b^4-6 a c d b^3+9 a^2 d^2 b^2\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (a d-b c)^2 f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

(Sqrt[c + d*Sin[e + f*x]]*(-(b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])^2) + (3*(2*a
*b^3*c*Cos[e + f*x] - 3*a^2*b^2*d*Cos[e + f*x] + b^4*d*Cos[e + f*x]))/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^2*(a + b
*Sin[e + f*x]))))/f + ((-2*(16*a^2*b^2*c^2 + 8*b^4*c^2 - 32*a^3*b*c*d + 2*a*b^3*c*d + 16*a^4*d^2 - 19*a^2*b^2*
d^2 + 9*b^4*d^2)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c +
d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*b^2*c*d + 4*b^4*c*d - 32*a^3*b*d^2 + 8*a*b^3*d^2)*Cos
[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a
*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d
)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e +
f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2
- 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-6*a*b^3*c*d + 9*a^2*b^2*d^2 - 3*b^4*d^
2)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*S
in[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c
+ d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c
+ d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[
e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin
[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[
-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*(a + b)^2*(-(b*c) + a*
d)^2*f)

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Maple [A]  time = 4.98, size = 867, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)
^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^2*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(-(-d*sin(f*x+e)-
c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b^3*c)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c
)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-
d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-3/4*b*d*(3*a
^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(
c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*
sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+1
/4*(15*a^4*d^2-20*a^3*b*c*d+8*a^2*b^2*c^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^4*d^2)/(a^3*d-a^2*b*c-a*b^2*
d+b^3*c)^2/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(
1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c
/d+a/b),((c-d)/(c+d))^(1/2)))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*sin(f*x + e) + a)^3*sqrt(d*sin(f*x + e) + c)), x)